Significance A central goal in invasion genomics is to identify and determine the mechanisms that underlie the successful colonization, establishment, and subsequent range expansion of invasive populations of nonindigenous species. Using a whole-genome approach, we evaluate the importance of genetic diversity for the successful establishment of nonindigenous species. Our study shows that genetic diversity per se is not the major factor driving invasions, since we observed all possible scenarios with invasive populations showing reduced, similar but also increased, genetic diversity relative to the native population. Using coalescent methods, we reconstruct the demographic history of the invasion and infer the source population of each invasion event, which shows that propagule pressure and multiple introductions play an important role in determining invasion success.